Master Solid Geometry: Conquer 3D Shapes in Mathematics Year 13

Question 1

A cube with a side length of 6 cm needs to be painted on all of its surfaces. What is the total surface area that needs to be painted?

Accepted Answer

144 cm²

Answer

36 cm²

Answer

72 cm²

Answer

216 cm²

Question 2

Rectangular prisms possess several defining properties. Which of the following is NOT among them?

Accepted Answer

All edges are equal in length

Answer

Opposite faces are congruent

Answer

Edges are perpendicular to each other

Answer

Faces are rectangles

Question 3

Given a pyramid with a square base and a side length of 5 cm, and a height of 10 cm, calculate its volume:

Accepted Answer

100 cm³

Answer

25 cm³

Answer

50 cm³

Answer

200 cm³

Question 4

Calculate the volume of a cone with a circular base having a radius of 3 cm and a slant height of 5 cm.

Accepted Answer

30π cm³

Answer

15π cm³

Answer

45π cm³

Answer

60π cm³

Question 5

Which formula is used to calculate the volume of a sphere?

Accepted Answer

V = (4/3)πr³

Answer

V = πr²h

Answer

V = Bh

Answer

V = πrℓ

Question 6

Select the statement that is NOT a property of a rectangular prism.

Accepted Answer

Edge lengths are all different.

Answer

Opposite faces are parallel and congruent.

Answer

All faces are rectangles.

Answer

Diagonals of all faces are congruent.

Question 7

Which property is NOT characteristic of a prism?

Accepted Answer

Curved surface

Answer

Parallel sides

Answer

Constant cross-section

Answer

Vertices

Question 8

Choose the formula for calculating the volume of a pyramid:

Accepted Answer

(1/3) * base area * height

Answer

base area * height

Answer

(1/2) * base area * height

Answer

(1/4) * base area * height

Question 9

Which shape constitutes the base of a cone?

Accepted Answer

Circle

Answer

Rectangle

Answer

Triangle

Answer

Square

Question 10

Choose the formula for calculating the volume of a cone:

Accepted Answer

(1/3) * base area * height

Answer

base area * height

Answer

(1/2) * base area * height

Answer

(1/4) * base area * height

Question 11

The volume (V) of a prism with base area A and height h is given by:

Accepted Answer

V = A * h

Answer

V = A * h / 2

Answer

V = 2 * A * h

Answer

V = A / h

Question 12

Which pyramid has a square base?

Accepted Answer

Regular pyramid

Answer

Oblique pyramid

Answer

Truncated pyramid

Answer

Triangular pyramid

Question 13

The volume (V) of a cone with radius r and height h is:

Accepted Answer

V = (π * r² * h) / 3

Answer

V = 2 * π * r² * h

Answer

V = π * r² * h

Answer

V = (4/3) * π * r³

Question 14

The surface area (SA) of a sphere with radius r is:

Accepted Answer

SA = 4 * π * r²

Answer

SA = 2 * π * r²

Answer

SA = π * r²

Answer

SA = 8 * π * r³

Question 15

The angle between the base and a lateral face of a regular hexagonal prism is:

Accepted Answer

60°

Answer

30°

Answer

45°

Answer

90°

Question 16

Identify the geometric property that does NOT apply to a prism.

Accepted Answer

Regular bases

Answer

Parallel bases

Answer

Equal lateral faces

Answer

Perpendicular lateral faces to bases

Question 17

Calculate the total surface area of a sphere with a radius of 7 cm.

Accepted Answer

616π cm²

Answer

308π cm²

Answer

154π cm²

Answer

77π cm²

Question 18

A cone has a slant height of 13 cm and a base radius of 5 cm. Calculate its height.

Accepted Answer

12 cm

Answer

5 cm

Answer

8 cm

Answer

10 cm

Question 19

Determine the radius of a sphere with a volume of 36π cm³.

Accepted Answer

6 cm

Answer

3 cm

Answer

9 cm

Answer

12 cm

Question 20

Determine the volume of a sphere with a 10 cm diameter.

Accepted Answer

500π cm³

Answer

(4/3)π cm³

Answer

250π cm³

Answer

1000π cm³

Question 21

Determine the lateral surface area of a cylinder with radius r and height h.

Accepted Answer

2πrh

Answer

2πr²

Answer

πr²h

Answer

4πr²

Question 22

Determine the equation of the sphere with center (2, -1, 3) and radius 5.

Accepted Answer

(x - 2)² + (y + 1)² + (z - 3)² = 25

Answer

(x - 2)² + (y + 1)² + (z - 3)² = 5

Answer

(x + 2)² + (y - 1)² + (z + 3)² = 25

Answer

(x + 2)² + (y - 1)² + (z + 3)² = 5

Question 23

A cone has a slant height of 10 centimeters and a base radius of 6 centimeters. Calculate the height of the cone.

Accepted Answer

8cm

Answer

6cm

Answer

4cm

Answer

2cm

Question 24

Calculate the surface area of a sphere with a radius of 5 cm. Express your answer in terms of π.

Accepted Answer

50π cm²

Answer

25π cm²

Answer

100π cm²

Answer

200π cm²

Question 25

Identify the regular polyhedron from the following options:

Accepted Answer

Cube

Answer

Triangular pyramid

Answer

Cylinder

Answer

Sphere

Question 26

Calculate the base area of a triangular prism with a base length of 6 cm and a height of 4 cm.

Accepted Answer

24 cm²

Answer

12 cm²

Answer

36 cm²

Answer

48 cm²

Question 27

Identify the net of a rectangular prism from the following options:

Accepted Answer

A rectangle

Answer

A square

Answer

A parallelogram

Answer

A trapezoid

Question 28

Calculate the total surface area of a cylinder with a radius of 3 cm and a height of 6 cm. Express your answer in terms of π.

Accepted Answer

54π cm²

Answer

18π cm²

Answer

36π cm²

Answer

72π cm²

Question 29

A cone has a radius of 3 cm and a height of 4 cm. What is its surface area?

Accepted Answer

24π cm²

Answer

30π cm²

Answer

18π cm²

Answer

12π cm²

Question 30

Which of the following shapes has the greatest surface area?

Accepted Answer

Sphere

Answer

Cylinder

Answer

Cone

Answer

Cube

Question 31

A tetrahedron has edges of length 5 cm. What is its volume?

Accepted Answer

6.25 cm³

Answer

25 cm³

Answer

12.5 cm³

Answer

3.125 cm³

Question 32

Which of the following is NOT a property of a sphere?

Accepted Answer

It has a constant height

Answer

Its volume is (4/3)πr³

Answer

Its surface area is 4πr²

Answer

Its radius is equal in all directions from the center

Question 33

A pyramid has a square base with a side length of 8 cm and a height of 12 cm. Calculate its volume.

Accepted Answer

256 cm³

Answer

1024 cm³

Answer

128 cm³

Answer

512 cm³

Question 34

Which of the following solids has a circular base?

Accepted Answer

Cone

Answer

Pyramid

Answer

Prism

Answer

Cube

Question 35

Which of the following is NOT considered a polyhedron?

Accepted Answer

Sphere

Answer

Pyramid

Answer

Prism

Answer

Cube

Question 36

Calculate the volume of a rectangular prism with a length of 6 cm, a width of 4 cm, and a height of 8 cm. Use the formula: V = lwh.

Accepted Answer

192 cm³

Answer

384 cm³

Answer

96 cm³

Question 37

Find the slant height of a square-based pyramid with a base side length of 10 cm and a height of 12 cm. Use the formula: l = √(h² + (s/2)²).

Accepted Answer

15 cm

Answer

25 cm

Answer

20 cm

Answer

10 cm

Question 38

Calculate the lateral surface area of a cone with a base radius of 6 cm and a slant height of 10 cm. Use the formula: A = πrl.

Accepted Answer

377 cm²

Answer

290 cm²

Answer

754 cm²

Answer

189 cm²

Question 39

Find the total surface area of a cube with an edge length of 5 cm. Use the formula: A = 6s².

Accepted Answer

150 cm²

Answer

200 cm²

Answer

100 cm²

Question 40

Find the slant height of a cone with a base radius of 5 cm and a height of 12 cm. Use the formula: l = √(r² + h²).

Accepted Answer

13 cm

Answer

10 cm

Answer

20 cm

Answer

15 cm

Question 41

A rectangular prism has dimensions of length 8 cm, width 5 cm, and height 3 cm. Calculate its surface area.

Accepted Answer

166 cm²

Answer

240 cm²

Answer

100 cm²

Answer

80 cm²

Question 42

A square-based pyramid has a base side length of 6 cm and a height of 4 cm. Determine its volume.

Accepted Answer

48 cm³

Answer

24 cm³

Answer

72 cm³

Answer

144 cm³

Question 43

Calculate the surface area of a sphere with a radius of 5 cm.

Accepted Answer

314.16 cm² (approximately)

Answer

628.32 cm² (approximately)

Answer

471.24 cm² (approximately)

Answer

157.08 cm² (approximately)

Question 44

A cone has a radius of 3 cm and a slant height of 5 cm. Calculate its lateral surface area.

Accepted Answer

47.12 cm² (approximately)

Answer

117.81 cm² (approximately)

Answer

28.27 cm² (approximately)

Answer

56.55 cm² (approximately)

Question 45

A cube has a volume of 64 cm³. Determine the length of one of its sides.

Accepted Answer

4 cm

Answer

2 cm

Answer

16 cm

Answer

8 cm

Question 46

Calculate the volume of a cylinder with a radius of 4 cm and a height of 10 cm.

Accepted Answer

502.65 cm³ (approximately)

Answer

251.33 cm³ (approximately)

Answer

125.66 cm³ (approximately)

Question 47

Which of the following solids has a constant cross-section across its length?

Accepted Answer

Prism

Answer

Pyramid

Answer

Cone

Answer

Sphere

Question 48

A right circular cone is sliced parallel to its base. What shape will the cross-section be?

Accepted Answer

Circle

Answer

Triangle

Answer

Rectangle

Answer

Ellipse

Question 49

A rectangular pyramid is cut by a plane parallel to its base. What shape will the cross-section be?

Accepted Answer

Rectangle

Answer

Trapezium

Answer

Triangle

Answer

Square

Question 50

A sphere is intersected by a plane passing through its center. What shape will the cross-section be?

Accepted Answer

Circle

Answer

Triangle

Answer

Ellipse

Answer

Semi-circle